well yes is the answer
records of NI payments ARE kept and I used to work in an office whose job was to count them up when someone retired and work out their pension
1968
I remember that someone aged 65 had never missed a weeks work because all the stamps - yes every one = was there !
Paper records
but that of course was fifty years ago but I can believe that the govt has failed to keep records as that means they have missed out on an opportunity to cut someones pension
I cant imagine that you can get anyone else's records besides your own unless you are an employer perhaps .....
and this is how you do it
https://www.gov.uk/check-national-insurance-record